## Interacting Particle Systems

*27 Jul 2023 12:03*

In the obvious sense, all of statistical mechanics is about "interacting particle systems". More technically, however, the name has come to refer to a class of spatio-temporal stochastic processes, in which time is continuous, space may or may not be discrete, and each spatial location can be in one of a discrete number of states --- interpreted as the number or type of particles at that point-instant. The global configuration evolves according to a Markov process. These are natural generalizations of cellular automata to continuous time, which raises some interesting mathematical issues, and adds a little more realism.

Standard CA update all cells synchronously, but changing this updating
scheme can change the qualitative behavior of a rule considerably.
(Fates and Morvan have a
nice paper on this, with a review of the published literature on the question,
which is a small slice of the unpublished folklore.) *Query*: When
synchronous and asynchronous updating in a discrete-time CA give very different
behaviors, which one matches the continuous-time interacting particle system?
This sounds like a question which could be resolved through
the usual Kurtz et
al. machinery for proving that a sequence of Markov processes converge by
manipulating their generators.

Particle filtering from state estimation goes here. The idea in that case is to represent possible hidden states of the system through a large but finite number of particles, located in the state space. In between observations, particles move independently, in accordance with the dynamics your model assumes on the state space. When observations are made, particles get re-sampled, with weights proportional to the likelihood of getting the current observation from the represented state. Particles at different locations (states) thus interact with each other through the population-averaged likelihood, rather than through the local interactions typical of physical models. Many people have noticed that this sounds like evolution, or at least a genetic algorithm....

See also: cellular automata; ergodic theory for Markov processes; filtering and state estimation; Markov models; mean-field games and mean-field control; non-equilibrium statistical mechanics; path integrals and Feynman diagrams for classical stochastic processes; pattern formation;

- Recommended:
- P. Del Moral and L. Miclo, "Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Nonlinear Filtering", in J. Azema, M. Emery, M. Ledoux and M. Yor (eds)., Séminaire de Probabilités XXXIV (Springer-Verlag, 2000), pp. 1--145 [Postscript preprint. Looks like a trial run for Del Moral's book.]
- Rick Durrett
- Stochastic Spatial Models: A Hyper-Tutorial
- Lectures Notes on Particle Systems and Percolation

- Bert Fristedt and Lawrence Gray, A Modern Approach to Probability Theory [Contains a good one-chapter account of the basics of interacting particle systems, but presumes knowledge of measure-theoretic probability and stochastic processes --- such as you'd get from reading the earlier chapters!]
- David Griffeath, Additive and Cancellative Interacting Particle Systems

- To read:
- David Aldous, "Interacting particle systems as stochastic social dynamics", Bernoulli
**19**(2013): 1122--1149 - E. Andjel, G. Maillard, T.S. Mountford, "A note on 'signed voter models'", arxiv:0709.3468
- Alexei Andreanov, Giulio Biroli, Jean-Philippe Bouchaud, and
Alexandre Lefevre, "Field theories and exact stochastic equations for
interacting particle
systems", Physical Review E
**74**(2006): 030101, cond-mat/0602307 - Chalee Asavathiratham, The influence model: a tractable representation for the dynamics of networked Markov chains
- Sven Banisch, Ricardo Lima, Tanya Araújo, "Agent Based Models and Opinion Dynamics as Markov Chains", arxiv:1108.1716
- Lamia Belhadji, "Ergodicity and hydrodynamic limits for an epidemic model", arxiv:0710.5185
- Vivek Borkar, Rajesh Sundaresan, "Asymptotics of the Invariant Measure in Mean Field Models with Jumps", arxiv:1107.4142
- Anne-Severine Boudou, Pietro Caputo, Paolo Dai Pra and Gustavo Posta, "Spectral gap estimates for interacting particle systems via a Bakry & Emery-type approach", math.PR/0505533
- Clive G. Bowsher, "Stochastic kinetic models: Dynamic independence,
modularity and
graphs", Annals
of Statistics
**38**(2010): 2242--2281 - Xavier Bressaud and Nicolas Fournier, "On the invariant distribution of a one-dimensional avalanche process", math.PR/0703750
- Amarjit Budhiraja, Paul Dupuis, Markus Fischer, "Large deviation properties of weakly interacting processes via weak convergence methods", Annals of Probability
**40**(2012): 74--102, arxiv:1009.6030 - Nicoletta Cancrini, Fabio Martinelli, Cyril Roberto, Cristina Toninelli, "Facilitated spin models: recent and new results", arxiv:0712.1934
- Sebastien Chambeu and Aline Kurtzmann, "Some particular self-interacting diffusions: Ergodic behaviour and almost sure convergence", Bernoulli
**17**(2011): 1348--1267 - Chan, From Markov Chains to Non-Equilibrium Particle Systems
- Leonardo Crochik and Tania Tome, "Entropy production in the
majority-vote model", Physical
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- Pierre Del Moral
- "Measure-Valued Processes and Interacting Particle
Systems. Application to Nonlinear Filtering Problems", The Annals of
Applied Probability
**8**(1998): 438--495 - Feynman-Kac Formulae
- Mean Field Simulation for Monte Carlo Integration

- "Measure-Valued Processes and Interacting Particle
Systems. Application to Nonlinear Filtering Problems", The Annals of
Applied Probability
- Pierre Del Moral and Arnaud Doucet, "Interacting Markov chain Monte Carlo methods for solving nonlinear measure-valued equations", Annals of Applied Probability
**20**(2010): 593--639 - Pierre Del Moral, Josselin Garnier, "Genealogical particle analysis of rare events", Annals of Applied Probability
**15**(2005): 2496--2534, arxiv:math/0602525 - Pierre Del Moral, Peng Hu and Liming Wu, "On the Concentration Properties of Interacting Particle Processes", Foundations and Trends in Machine Learning
**3**(2012): 225--389, arxiv:1107.1948 - Paul Doukhan, Gabriel Lang, Sana Louhichi, Bernard Ycart, "A functional central limit theorem for interacting particle systems on transitive graphs", math-ph/0509041
- Andreas Eibeck and Wolfgang Wagner, "Stochastic Interacting
Particle Systems and Nonlinear Kinetic
Equations", Annals of
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**13**(2003): 845--889 - Alison M. Etheridge, An Introduction to Superprocesses
- Joaquin Fontbona, Helene Guerin, Sylvie Meleard, "Measurability of optimal transportation and convergence rate for Landau type interacting particle systems", math.PR/0703432
- Henryk Fuks and Nino Boccara, "Convergence to equilibrium in a class of interacting particle systems evolving in discrete time," nlin.CG/0101037
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**138**(2010): 476--495 - Thierry Gobron and Ellen Saada, "Coupling, Attractiveness and Hydrodynamics for Conservative Particle Systems", arxiv:0903.0316
- A. Greven, F. den Hollander, "Phase transitions for the long-time behaviour of interacting diffusions", math.PR/0611141
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- Jane Hillston, a href="http://cambridge.org/9780521571890">A Compositional Approach to Performance Modelling
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**126**(2007): 585-642 - Nicholas Lanicher, "The Axelrod model for the dissemination of culture revisited", Annals of Applied Probability
**22**(2012): 860--880 - Julio Largo, Piero Tartaglia, Francesco Sciortino, "Effective non-additive pair potential for lock-and-key interacting particles: the role of the limited valence", cond-mat/0703383
- Alexandre Lefevre, Giulio Biroli, "Dynamics of interacting particle systems: stochastic process and field theory", arxiv:0709.1325
- Thomas M. Liggett
- Interacting Particle Systems
- Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes
- "Stochastic models for large interacting systems and
related correlation
inequalities", Proceedings
of the National Academy of Sciences (USA)
**107**(16413--16419)

- E. Locherbach, "Likelihood Ratio Processes for Markovian Particle
Systems with Killing and Jumps", Statistical Inference for Stochastic
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**5**(2002): 153--177 - Nadia Loy, Andrea Tosin, "Markov jump processes and collision-like models in the kinetic description of multi-agent systems",
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**18**(2020): 1539--1568, arxiv:1905.11343 - Frank Redig, Florian Völlering, "Concentration of Additive Functionals for Markov Processes and Applications to Interacting Particle Systems", arxiv:1003.0006
- Daniel Remenik, "Limit Theorems for Individual-Based Models in Economics and Finance", arxiv:0810.2813
- David Schnoerr, Ramon Grima, Guido Sanguinetti, "Cox process representation and inference for stochastic reaction-diffusion processes",
Nature Communications
**7**(2016): 11729, arxiv:1601.01972 - A. V. Skorohod, Stochastic Equations for Complex Systems [chapter 2 being "Randomly Interacting Systems of Particles"]
- Anja Sturm and Jan Swart, "Voter models with heterozygosity selection", math.PR/0701555
- Denis Villemonais, "Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift", arxiv:1005.1530
- Biao Wu, "Interacting Agent Feedback Finance Model", math.PR/0703827